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Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut

Authors Pasin Manurangsi, Luca Trevisan

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ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Approximation algorithms
  • Exponential-time algorithms
  • Vertex Cover
  • Sparsest Cut
  • Balanced Separator

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Abstract

In this work, we study the trade-off between the running time of approximation algorithms and their approximation guarantees. By leveraging a structure of the "hard" instances of the Arora-Rao-Vazirani lemma [Sanjeev Arora et al., 2009; James R. Lee, 2005], we show that the Sum-of-Squares hierarchy can be adapted to provide "fast", but still exponential time, approximation algorithms for several problems in the regime where they are believed to be NP-hard. Specifically, our framework yields the following algorithms; here n denote the number of vertices of the graph and r can be any positive real number greater than 1 (possibly depending on n). 
- A (2 - 1/(O(r)))-approximation algorithm for Vertex Cover that runs in exp (n/(2^{r^2)})n^{O(1)} time. 
- An O(r)-approximation algorithms for Uniform Sparsest Cut and Balanced Separator that runs in exp (n/(2^{r^2)})n^{O(1)} time. 
 Our algorithm for Vertex Cover improves upon Bansal et al.'s algorithm [Nikhil Bansal et al., 2017] which achieves (2 - 1/(O(r)))-approximation in time exp (n/(r^r))n^{O(1)}. For Uniform Sparsest Cut and Balanced Separator, our algorithms improve upon O(r)-approximation exp (n/(2^r))n^{O(1)}-time algorithms that follow from a work of Charikar et al. [Moses Charikar et al., 2010].

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Pasin Manurangsi and Luca Trevisan. Mildly Exponential Time Approximation Algorithms for Vertex Cover, Balanced Separator and Uniform Sparsest Cut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.20

Author Details

Pasin Manurangsi
  • University of California, Berkeley, USA
Luca Trevisan
  • University of California, Berkeley, USA

Funding

This material is based upon work supported by the National Science under Grants No. CCF 1655215 and CCF 1815434.

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